Introduction to Chalker- Coddington Network Model Jun Ho Son

Hall Effect Classically explained, Lorentz force Linear relationship between current and Hall voltage

Quantum Hall Effect Low Temperature/high magnetic field

Quantum Hall effect

Why is Hall Conductance Quantized? Quantum mechanics needed, but no interaction Laughlin’s argument – Gauge Invariance TKNN Invariant (Integer) -> Associate chern number to the quantum number.

Protected Edge Mode Chiral edge state Absence of backscattering, robust against disorder Semi-classical “skipping orbits”

Phase Transition in Quantum Hall System Plateau Only have peak at the transition

Anderson Localization and Quantum Hall Phase Transition E DOS disorder E DOS e Localized state at edges, extended state at centers -> Insulator at plateau, finite conductance at crossing extended localized

Chalker-Coddington Network Model Simplified model for disordered Integer Quantum Hall transition Quantum Hall transition as tunneling/scattering and percolation

Motivation : tunneling between electron orbits around equipotential line Alternative treatment: 1D edge modes and scattering between two edge modes Chalker-Coddington Network Model Chalker and Coddington (1988)

Simplify : square electron orbits arranged in checkerboard pattern Scattering at the vertices Disorder introduced by randomness in phase gain between two vertices Chalker-Coddington Network Model

Build quasi-1D geometry –narrow, long strip (usually 10^5~10^6) Build a transfer matrix that describes electron propagation along the strip or the prism Transfer Matrix Approach T T T T T

Main Interest : How fast 1D wavefunction decays out along one-dimensional direction? –Insulator : decays quickly –Metal : extended through the length. Lyapunov exponent of transfer matrices characterize this “decay speed” Transfer Matrix Approach Metal Insulator Length Wavefunction Magnitude

Transfer Matrix Approach

QR Decomposition

Behavior Insulator

Critical Exponent

Why is this model interesting? Conceptual interest – Contains all qualitative/some quantitative information Critical Exponent Topological Invariant Fulga et al (2012) – Analytical Features : Mapping to Dirac equation Ho and Chalker (1995 ), field-theoretic model Ryu et al (2009) Numerically Cheap – Brute-force diagonalization is often bad……

Quantum Spin Hall Effect

Topological Insulator Such “topological phases” can occur under different dimensions/symmetries Ryu et al. (2010) Quantum Hall System Quantum Spin Hall System

Network Model of 2D Topological Insulators Add additional degrees of freedom, constrain random phase/scattering matrix to obey the symmetry Obuse et al(2007), Fulga et al(2012) Obuse et al(2007),

Network Model of 2D Topological Insulators Insulator-Metal- insulator transition instead of Insulator- Insulator transition Obuse et al(2007)0

3D Network Model

3D Extension? – Stacked 2D network model  Weak topological insulator Chalker and Dohmen(1995), Obuse et al(2014) – Special quantum-classical mapping to study the model that belongs to class C Ortuno et al(2009) – Another approach?

Surface states of topological insulator -> three 1D orbits on sphere –Scattering between orbits within sphere Can simplify further as cubic geometry 3D Network Model Construction

Stacking clusters in face-centered cubic lattice Cutting out the hexagonal prism -> Quasi- 1D geometry 3D Network Model Construction

Inter-cluster nodes and intra-cluster nodes –Inter-cluster nodes : parameterized by a single parameter, similar to 2D model –Freedom to choose intra-cluster nodes, in contrast with 2D network model 3D Network Model Construction

Result Insulator Metal Mobility edge

Conclusion Chalker-Coddington model gives numerically cheap method to understand disordered quantum hall transition The model can be extended to 2D topological insulator Extension to 3D?